On the role of semismoothness in nonsmooth numerical analysis: Theory
Abstract
For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth multifunctions.
Cite
@article{arxiv.2405.14637,
title = {On the role of semismoothness in nonsmooth numerical analysis: Theory},
author = {H. Gfrerer and J. V. Outrata},
journal= {arXiv preprint arXiv:2405.14637},
year = {2026}
}